Energy Partition 



An analysis of the oscillations of a non-migrating bubble by Arons and Yennie 

 [4] has shown that despite the fact that the bubble remains spherical a strong energy 

 dissipation takes place near the bubble minima. For instance, at the first bubble 

 minimum 25 per cent of the original bubble energy is acoustically radiated by the 

 bubble pulse. Another 37.5 per cent is dissipated by a mechanism which is not clearly 

 understood at the present time. This leaves only 37.5 per cent of the original bubble 

 energy for the next pulsation. This is illustrated in Fig. 8. 



This energy dissipation of a non-migrating bubble is somewhat mysterious. It 

 is not caused by a collision of the bubble surfaces as we have seen it for the migrating 

 bubble. It may have something to do with the stability of the gas- water interface. 

 According to G. I. Taylor's [16] criterion, this interface is highly unstable near the 

 bubble minimum, but is stable for the major portion of the oscillation. This unstability 

 may produce a spray of water which is projected into the compressed gas and reduces 

 the internal energy of the gas by cooling. 



For migrating bubbles [14] the energy radiated by the bubble pulse decreases 

 with increasing migration Fig. 8. The amount of the dissipated energy (called 

 "unaccounted for" in the figure) remains about the same, although the mechanism 

 probably changes. Therefore, the bubble energy (of the second cycle) increases with 

 increasing bubble migration as shown in Fig. 8. 



Theoretical Treatment for Non-Migrating Bubbles 



There is a great number of publications on the theory of the oscillating gas 

 bubble [1, 15]. Most of these assume that the bubble pulsates in an incompressible 

 medium. Since the bubble pressure is low for a great portion of the oscillation, such 

 approximate theories describe the movements of the bubble very well, except for short 

 moments near the bubble minimum. Of course, these theories cannot describe the 

 early bubble expansion shortly after the detonation and the energy losses at the minima. 

 An acoustic approximation or improved version of it can account for the energy 

 radiation by the bubble pulse, but not for the additional energy dissipation at the 

 minimum. Therefore, all these calculations are applicable only to one cycle of the 

 oscillation. For each of these cycles, the parameters of the theory must be redetermined 

 on the basis of the empirical evidence on the energy redistribution at the bubble 

 minimum. It is possible to adjust the parameters of the acoustic theory in such a 

 way that the observed radius-time curve is reproduced over several cycles. Such cal- 

 culations would yield too large amplitudes for the bubble pulse and, therefore, are in 

 poor agreement with pressure measurements. 



Migration Caused by Bounding Surfaces 



Besides gravity other factors may produce a migration of the bubble. For 

 instance, a rigid surface attracts the pulsating bubble and a free surface such as the 

 water surface repells it. This phenomenon is analogous to the classical Bjerknes 

 spheres. Bjerknes demonstrated long ago through the use of air-filled rubber bags in 

 water that bubbles attract each other when their oscillation is in phase and that they 

 repel each other when pulsating out of phase. A bubble that oscillates near a rigid 

 surface can be represented by the bubble and its image pulsating in phase. Therefore, 

 the bubble and its image attract each other. At a free surface, the image of a sink 

 is a source and vice versa. Hence, the image of the bubble pulsates out of phase and 

 the bubble is repelled from the free water surface. 



For an explosion in shallow water these two effects oppose the upward migra- 



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